PETER: Welcome to

Authors at Google and welcome to Alex Bellos. I don’t know if any of

you saw his earlier talk a couple of years ago around

his first book, “Adventures in Numberland,”

which was a huge hit. And on a personal business,

it was a huge hit for me because my 10-year-old

son at the time I couldn’t get him to

read any books at all. And in desperation,

we were on holiday, and I kind of handed him

“Adventures in Numberland” and said have a look at this. And he quickly became

extremely hooked on and spent the entire holiday

going yam, tan, tethera which I’m sure you’ll know is the

ancient English way of counting sheep. And when you get to

15, it’s bumfit I think I’m right in saying. And he spent the entire

holiday talking about bumfit. So it sort of both hooked

him in an interest in maths and gave him a license

to use bad language so it was kind of perfect. So the new book is “Alex

Through the Looking-glass.” Alex is going to

talk for half an hour or so and then take

some questions? ALEX BELLOS: A little bit

more than that but, because I want to be very informal,

so if there’s anything you don’t understand or you

want more, just shout out, put your hand up. So let’s kind of have

the kind of Q and A as we go I think because

we’re a little bit delayed. I’m going to have to– PETER: And I’ll hand

if over [INAUDIBLE]. Keep going. Cheers. ALEX BELLOS: Usually,

I’m used to giving talks to book festivals where the

kind of mathematical numeracy is going to be a lot lower

than you guys are here so I might whip through

it but you might have slightly more

high-brow questions. So thank you very much, Peter. I’m Alex Bellos. This is me. Oh, my god. I just pressed something. That’s me with a weird

stain on my red jumper. I don’t know why

I was given that. And this is my new book, “Alex

Through the Looking-glass,” which sounds very much like it’s

a sequel to “Alex’s Adventures in Numberland” but

it isn’t really. Both can be read separately. You don’t need to have read

the first one to read this. If there is a

difference, the first one was just an exploration about

mathematical abstraction and what we get when we use it. This is a little bit

more kind of applied. It’s about using

abstraction to learn about the world, mathematics

being the language that we use to talk about the

world, and also how life reflects numbers

and numbers reflect life. I also wanted to get into

kind of psychology of numbers and interesting

psychological responses that we have to numbers which

are quite surprising I think and definitely under-researched. Now, how do you write a

book on maths and numbers and make it interesting. It’s quite difficult because

the nature of the subject is dry and sometimes it’s

conceptually challenging. So I’ve got a degree

in maths and philosophy and then I became a journalist. So my approach is

always to tell stories. And I think that

by telling stories, if you can interest

someone in the story, then you don’t need

to say, oh, by the way I’m going to teach

you some maths. You just learn the

maths on the way. And where do you

look for stories? Well, the kind of classic

place to look for a story is in oneself. And so, actually, my story as

a popularizer of mathematics is really where

this book begins. And I have to say I don’t

think it happened at my talk at Google last time, but pretty

much every talk that I did when the first book came out,

I would give what I thought was a fantastically fascinating

talk about maths and numbers. And someone would put

their hand at the end of it and they would say, yes, sir. And they would just say, so

what’s your favorite number? And I can just

remember thinking, oh, have you learned nothing? Who has a favorite number? I’m grown-up. You’re a grown-up. And I would just think they

were kind of teasing me. It was way of belittling

the subject on myself. Until I kept on being

asked this question. I would give talks

at universities, secondary schools,

books festivals. People were always asking it. So once I just said,

oh, I didn’t know. What’s your favorite number? But rather than saying,

ah, they went, oh, 12, gave me a considered response. And I just was like, what? And then the person next

to them said, oh, no, my favorite number is 13. I said, well, who here’s

got favorite numbers? And at least half the

audience of grown-ups put their hands up. And this happened

a few more times. So I thought, well,

this is interesting. From being something that I

thought was completely idiotic, I thought, well, if

grown-up people who are literate and numerate

have serious [INAUDIBLE] emotional reactions

numbers, this is interesting and why don’t I try

and research it? And so what I did is I thought

I would undergo a survey to try and find the

world’s favorite number. But essentially it was

to try and quantify emotional responses to numbers. So I set up this website

here, favouritenumber.net. I realized it was going to

have to be international, so I also bought this

one, favoritenumber.net, just so no one was excluded. And within a few

weeks– at that time, four years ago, I maybe had

only a few thousand followers on Twitter. I just put it on Twitter. What’s your favorite number? Join my survey. Within a few weeks,

more than 30,000 people from around the

world had replied. And so this was already

vindicating this idea. This is something

that serious people feel passionately about it. Now, what were the results? The results were this. 7, 3, 8, 4, 5, 13, 9, 6, 2, 11. Which “Metro” said was

the most confusing top 10 list ever written. Which it is. Now, if you go back

to favoritenumber.net, I have all the results up there. There’s a big Excel file of all. There’s more than 1,000

individual numbers and all their frequencies. And you can get

them there if you are interested to do

an analysis on them, if you have a statistical mind. But what was interesting

was that when I launched the results, it

was an actual news story. Again, showing that people

are interested in this stuff. It was a news story

in the “Times.” It was a news story in the

“Daily Mail” of all places. But it also started

going around the world. This is “Glamour”

magazine in Paris. And I don’t think I

found one country that didn’t report it in some way. Just going to mention

a few of them. Hungary. Hungary has a huge brilliant

mathematical tradition. And per capita, I think

that more Hungarians entered the survey than

any other nation, so they were really

fascinated by it. Obviously, it was

[INAUDIBLE] Greek, Greece being the beginnings

of Western thought. By the time, it got

to India, I was now a Brazilian mathematician. But India– there

are certain countries where mathematics and the

idea of doing arithmetic is really an important

part of national identity. And so it was all over the

regional press in India. Vietnam, and obviously

I can’t read Vietnamese. And it just became

quite interesting what happens in the different

photo archives countries when you put in seven. This is what you get

when you’re in Vietnam. And when you’re in China,

this is what you get. So it was a new story. But what is the

moment that you really know that it has touched

the public consciousness, it’s become a real news story? It’s when you’re a

question– volume, please? Three days afterwards, it was

on “Have I Got News for You.” [VIDEO PLAYBACK] -Alex Bellos asked 44,000

people to submit their favorite number– ALEX BELLOS: So yeah there

were 44,000 people in the end. But I only did the cleaning

of the data for 33,000. The Nigel [INAUDIBLE] numbers. —top ten world’s

favorite numbers. -Yes. -Yeah. ALEX BELLOS: The

reason why this is funny is because we actually

do have feeling– things are only fun if there’s an element

of something serious about it. -Number ten, the tenth most

popular is, in fact, 11. ALEX BELLOS: [INAUDIBLE]

mathematical joke. -14. -No. -21? -No. At number nine is

the number two. The most popular number

in eighth position is six. And seven, the seventh most

popular number is nine. And the sixth most

popular number is 13. OK. Here’s the top five. -This is going to be on Channel

Four for a whole evening. There’ll be some talking

heads in a minute. What did you of six? Eh, not it. ALEX BELLOS: This is

actually my idea– -I’ve always liked five because

it’s a working class number. That’s what I like about five. But I think that

three [INAUDIBLE] and I thought three was a

magnificent num– oh, I’m sorry. I’ll stop auditioning. Sorry. Where am I? -Let’s firstly

complete the top five. The number five is five. And number four is four. And number three is eight. And number two is

three, the magic number. And, of course, seven is

the top one, number one. [END VIDEO PLAYBACK] ALEX BELLOS: OK. It’s interesting that

everyone laughed at five being five and four being

four because, essentially, we like small numbers. And the bigger the numbers

get, we like them least. This is what the survey says. So a number should, if there

are no other considerations punch more or less exactly

their numerical value. So those are almost the ones

that are the least funny or the least interesting. And what happens

and what you see is the odd numbers and

especially prime numbers are way up the rankings

and round numbers really, really way back

down the rankings. We do not really

like round numbers, but we really like

prime numbers. And that is

something which, even though the quantitative research

you can dismiss and say, oh, this is silly because

it’s 44,000 people but it was 44,000 people who

already feel passionately about it who selected

themselves to join the survey. Even amongst them, there

are very coherent patterns of these emotional responses. So on the survey, I said,

what’s your favorite number? But I also said, why? I just left it blank. There’s a little space. You could write it in there. And to me, it was the

qualitative results of the survey that were

actually a bit more interesting and a bit more

surprising and that felt a bit like

proper new research. So I’m going to just take you

through some of the entries with some of the reasons. And we start to

understand what it is about numbers that make

people excited by them. So this number was

selected, 10 over root two. Now, in this

audience, does anyone know what 10 over root

two is in decimals? It’s 10 times 1 over root two. Which if you work out is

actually square root of 50. 7.07 which in America is

called the Boeing number because of 707. And I had originally thought

that Boeing was called 707 just because this looks a

little bit like a plane, two wings and a fuselage. Apparently that’s

not the reason why. But you will find that certain

numbers work in brands really well based on just how they

look and how they sound. Because even though

numbers are supposedly these abstract entities

just representing quantity and order. How do we get to them? They exist within

culture, within language. They have to look like

something as a symbol. And all these things

actually influence how we understand even

the numerical properties of numbers. I’m going to read you the

reason why the person chose 707. It wasn’t because they were

an aviation enthusiast. People wrote down their

nationalities, their genders, their level of mathematical

ability, and their age. And this was from a

woman in Canada, aged 34, with university-level maths. She said, this number

is 10 root two. I’d been doing a

lot of trig homework for a calculus class I

was taking in undergrad and this number appears a lot. At the time, I was sort

of weirded out by the fact that I kept waking up at 7:07

in the morning instead of 7:30 when my alarm was set for. Anyway, one Saturday, I went

to my local art supply store and bought some paintbrushes. To my surprise, the

total came to $7.07. And I sort of blurted out,

oh, that’s 10 over root two all over again. So the very cute

cashier, for whom I had a rather pathetic crush

and who I was constantly embarrassing in front of,

after explaining myself, he was duly impressed and

began embarrassing himself in front of me whenever

I came to the store. And from that point

on, I realized there’s a brand of arty guys

that like nerdy girls and this still makes me happy

some 15 years later. OK. This is written in a tiny

little white space on a website. Just the story, the passion. There’s so much stuff

that you could analyze. Also the fact that when you

start to think emotionally about a number you

just see it absolutely everywhere even though numbers

don’t really predominant one over the other in

such a strong way. 1,000,000,007. Why was this chosen? This was chosen by

21-year-old male Russian because it’s the largest

prime number I can remember. So the fact that people take to

remembering large prime number. A prime number I’m sure you now. A prime number is a number that

only divides by itself and one. So they start with two. Two is the only even

prime number, then it goes 3, 5, 7, 11,

13, 17, 19, et cetera. Now, 1,000,000,007

is interesting because 1,000,000,009

is also a prime number. In fact, you get this things

called twin primes which are prime numbers

that are two apart. And it’s conjectured that there

are an infinite number of them. But the largest one

that’s know of the form one, loads of zeroes, then

seven, one, loads of zeroes, then nine is 1,000,000,007

and 1,000,000,009. So you can kind of

think you’re really clever to remember

[INAUDIBLE] prime number. It would be clever if it

was the second highest prime number I can remember. 219. Now, this was chosen by

an 18-year-old Irishman because it’s the lowest

whole number that does not have its own Wikipedia entry. So it’s too boring. It’s the lowest boring

number, basically. Every number from one to 218

has it’s own Wikipedia entry. And, actually, this was

a couple years ago now. So I’ve looked. There’s an inflation in

lowest boring numbers. And now the lowest

boring number is 224. OK. But you might not

think that Wikipedia is a very good judge of what

makes a number interesting or not. And so I thought I would look

in the online encyclopedia of integer sequences, which is

a kind of Bible of mathematics and really of all science. It contains about 250,000

separate sequences. And a sequence is just

a list of numbers. They can be infinite. They can be finite. So the prime numbers,

that’s a sequence. The square numbers, at

one, four, nine, sixteen, twenty-five. That’s a sequence. What it be interesting

to find out what is the lowest number

that has never appeared in the online encyclopedia

of digit sequences, which you could probably say is the

only number– well, probably the lowest number never

to have to appeared in any mathematical

piece of research in the last 3,000 years

since, really, there’s been mathematics. And that number

is 14,228, which I thought was quite

low personally. But it’s also interesting. It’s very, very

divisible, isn’t it? Divided by two obviously. This idea that numbers

that are really interesting are the prime numbers,

the odd numbers. Finally, this number 11,

which is also a cheap plug because I also

used to live in Brazil and wrote a book on Brazilian

football, which has also been updated for the World Cup. So if you’re interested

in finding out about Brazil and the World

Cup buy my book about it. It has [INAUDIBLE], who

is the– well, he’s not the captain, actually,

at the moment, but he’s the talisman of

the Brazilian national team. And the reason why 11 was

chosen was from an American, 36 years old, and I like to

think that he was from the deep South because if he wasn’t, it

doesn’t really make much sense because he said, I like 11

because it sounds like lovin’. So, why? What are the reasons

why seven was the world’s favorite number? I made a little animation,

hoping the sound is OK. [VIDEO PLAYBACK] -We’ve been obsessed by the

number seven for as long as we know. Go back to the

earliest writings there is, on Babylonian clay

tablets, and they’re just full of sevens. Then you have seven dwarves,

seven sins, seven seas, seven sisters. The list just goes on and on. So why is seven so special? One argument is that

there are seven planets in the sky visible

to the naked eye. To me, that’s just coincidence. There’s a much more

compelling reason. Seven is the only number

among those that we can count on our hands– that’s

those from one to 10– that cannot be divided or

multiplied within the group. So one, two, three, four, and

five, you can double them. six, eight and 10,

you can half them. And nine you can

divide by three. Seven is the only

one that remains. It’s unique. It’s a loner, the outsider. And humans interpret this

arithmetical property in cultural ways. By associating seven

with a group of things, you kind of makes

them special too. The point here is

that we’re always sensitive to

arithmetical patterns and this influences our

behavior even if we’re not conscious of it and irrespective

of our ability at maths. [END VIDEO PLAYBACK] ALEX BELLOS: There’s

another thing which kind of ties into this. If you were to ask someone– and

this is a psychology experiment that’s been repeated many

times– just think of a number off the top of your

head between one and 10, most people say seven. And why they say seven,

essentially– again, you’re doing mathematics

without realizing. You think, well, I’m

not going to say one because that’s not off

the top of my head. That’s just too obvious. It’s not sort of

arbitrary enough. I’m not going to say 10. I’m not going to say five. Well, you’ve just done

the two times table. And then you sort of say,

well, two, that’s also. I wouldn’t choose two. Then you eliminate

two, four, six, eight. You do exactly what

I was doing in there and you’re left with seven. Seven– it’s the most difficult. It’s the one that’s

[INAUDIBLE] to stretch where it feels the most random. Now, if we go back to

the qualitative results of the favorite number survey,

I was trying to work out, is there any way that I

could grasp or explain the kind of collective

views about numbers? And I was thinking,

well, how can I do this. And I was reading all this

more than 30,000 reasons why people like numbers. And I noticed that the language

used to describe each number was different. So I’m going to

pluck the adjectives that we use to describe

different numbers. Number one was

described– this is each word in a

different reason– as independent, strong,

honest, brave, straightforward, pioneering, competitive,

dramatic, egoistic, hot-headed, and lonely. So imagine we were a

creative writing class. And I said this is like the

main protagonist of our story. Those are really quite coherent

personality traits there. Two. These are words used to

describe the number two. In fact, sorry. That was number one. Two. So look at number two and see

if you visualize these traits. Simple, elegant, cautious, wise,

observant, pretty, fragile, open, sympathetic,

comfortable, flexible, subtle, inconspicuous. This is from lots

of different people and they’ve got lots

of different reasons but together, again, there

is a kind of coherence there. And what struck me most of

all was that if these were a creative writing

class and we were having to imagine characters for

the numbers one and two, one is really male

characteristics or traditionally

male characteristics. And two, traditionally

female characteristics. So the cartoonist of the

book did exactly that. This is how she interpreted it. We could do a fantastic kind

of rom-com or romantic story with one as the man

and two as the woman. Now, I saw this and

I thought, well, this is really quite striking. But the more that I read–

and I read quite a lot of stuff on the history

of maths and the growth of intellectual ideas and

also about psychology– it’s really quite

common, [INAUDIBLE]. So numbers were invented

about 8.000 years ago, say, in Sumeria, which

is present-day Iraq. The very first

words that we know that we used by humans for

the number one and the number two– the word

for the number one is the same words as

the word for phallus and the word for two was

the same word for woman. So right from the beginning,

there– for whatever reasons. We don’t know. You can imagine, I suppose. One is male and two is female. Actually, a friend of mine who’s

a quite well known novelist, she said well, Alex, I

was reading your stuff about masculine and feminine. And I just thought that

everyone thought that. I’ve always thought– and she’s

one of the smartest people that I know. And she was saying, yeah,

well because it’s obvious that one and odd numbers

are masculine and two and even numbers are feminine. They obviously are because two

you can kind of split apart. They can become two new things. It’s like they

kind of give birth. And obviously, I had

never thought that at all. But the fact that

someone– there are people who are

smart who think these. We do relate to numbers

in emotional ways. Pythagoras who’s the

beginning of Greek mathematics had this Pythagorean

brotherhood. And they believed that

odd numbers are masculine and even numbers are feminine. And I discovered

some research that had been done quite recently

in which respondents or participants were asked

to look at babies who are six weeks old of

indeterminate gender and all you have to do is

just say, is this a boy or is this a girl. Always, they had numbers

at the bottom of them. But the people were told. Don’t look at the numbers. There’s going to be a number. But just don’t look at it,

just look at the child. And what the

experimenters were doing is that sometimes it was all

three even numbers, sometimes all three all numbers,

all kind of randomized, proper science and that. And it turns out that,

with odd numbers, you’re 10% more likely

to think that a baby of indeterminate gender is male

and with even numbers, 10% more likely that the baby is female. So there is still this kind

of subconscious association with odd numbers with

masculinity and even numbers with femininity. Also, when I [INAUDIBLE] started

to get interested in this idea that numbers have personalities. And even though if I had

been talking to myself a few years ago I

would have thought that absolutely ridiculous. When I started to

think about it, I though, yeah, well I

use numbers all the time. Yeah, certain numbers do kind of

feel a bit different to others. And then when you

start looking at even Shakespeare noticed this. This is from “Merry

Wives of Windsor,” Falstaff, They say there

is divinity in odd numbers, either in nativity,

chance or death. And it’s true. All our most mystical

religious symbolic numbers are odd numbers in

the Western tradition. So three, Christianity,

seven, 13, five a little bit. They’re all low primes

and this is interesting. And there is

psychological research, proper psychological

research, not just my online survey,

that starts to give us a bit of an inkling about

why this might be the case. So one of the experiments was

you’re just shown two digits. Either they’re both odd

or they’re both even or one is even and one is odd. All you need to do is

just press a button when they’re both the same. So either when they’re both

even or when they’re both odd. Turns out you’re much slower

pressing the button when they’re both odd and

you make more mistakes. In other words, it kind of takes

us longer just to process it, just to think about them,

not to do any math with them, just to think about odd numbers

than it does with even numbers. And this sort of gives this

idea that somehow there’s sort of more room,

there’s more space, to have a kind of emotional

interaction with them. The final experiment I

want to talk about in terms of these psychological and

emotional responses to numbers is all about branding. And I’m sure we’re going to

be seeing a lot more of this because this is

quite recent work. The scientists, the

academics, wanted to know if we are more

likely to be attracted to want to buy and pay

more money for brands if they have different

types of numbers on them. So WD-40 is a very

popular brand. That’s an even number. Oxy10. That’s an even number. There’s something sensible,

safe about even numbers. And it turns out that, for

household products, for things that you want to

work and be reliable, people are much more

likely, will pay more money, if they have an

even number on them. So the experiment here was

two packs of contact lens, one is the brand called

Solus36, one is Solus37. They’re absolutely

exactly the same. You do all these tests about

how to evaluate the brand. Solus36 is much more popular. People pay more money for it. They want it. They like it better. OK. So this is the first

of the experiment. Then they add in this

tagline, 6 colors. 6 fits. Just to see if this affected how

much you wanted the brand how much you liked it,

how much money you were prepared to pay for it. And what happened

is that Solus36 becomes even more liked

and Solus37, which wasn’t liked very much

in the first place, becomes even less liked. And the argument put

forward, the hypothesis, by the scientists is that the

reason why we like Solus36 is that we– how do

we learn numbers? We learn numbers by

learning times tables by rote when we’re children. So we are much more

familiar with numbers that are in times tables. So 36. Whenever we do the six

times tables, we get it. We do the 12 times

tables, we get to it. And what happens? We never get to 37 because

it’s a prime number. It’s not on any times table. It will be the answer to very

few– basically, as a child, you will never utter or

think about the word 37 but you will use

36 all the time. So this idea that it’s

what you’re familiar with is that fluency of processing. And you misattribute this

fluency of processing for liking of the product. So it kind of feels good. Oh, I like that. Oh, it’s like a friend,

a friendly product. And this is what’s

exacerbated by the tagline because subconsciously

we’re seeing six, six, 36, and it’s like, oh, yeah. It’s like a nursery

rhyme from school. I remember that. I like that. I really like those

contact lenses. And with six, six, 37, there’s a

kind of mathematical cacophony. We don’t like it. So that brand is

really unattractive. And they did a lot

more experiments. And it’s interesting that when

you have an ad for something and if you subtly put numbers

in it that relate arithmetically together, you can

increase quite drastically the liking for that brand. So in the future

I’m sure there’ll be lots of numbers subtly

positioned within ads. Which you tend to

think, oh, god, I hated learning maths by rote. I hated learning

my times tables, but actually that either can

be kind of leveraged to make you feel good about a product

is ironic to say the least. That is essentially what is

the beginning of my book, the story about me and

the favorite numbers and emotional

connections to numbers which is how life

reflects numbers. The rest of the book,

so the majority of it, is kind of proper

more serious maths. And I’m going to just talk to

you about one part of it now that begins with– we’re

talking about prime number. So let’s just repeat

prime numbers. These are the prime numbers. The numbers that only divide

by themselves and one. Those are the first seven–

I think, yeah. –of them two, three, five, seven,

eleven, thirteen, seventeen. We live in the age of algorithms

as we’re told every day. What was probably the

very first algorithm in mathematics and,

therefore, in civilization? It was probably what’s called

the sieve of Eratosthenes which was the algorithm

to try and find all the prime numbers

under a certain number. So this is a grid of the

numbers from one to 100. I’ve done them in six rows. We’re going to use the

sieve of Eratosthenes to try and find all

the prime numbers. OK. What do you– you start at the

bottom, you get to a number, if the number has

not been eliminated, it’s a prime number. And then you eliminate

all multiples of it. So let’s start with one. One’s not a prime number

because we start at two. Get to two. Then we eliminate

all multiples of it which is basically

all even numbers. And it’s nice when you

do it in rows of six. So next one. We get to three. Three’s a prime number. Then we eliminate all

multiples of three which is just one more line

because the bottom row is all multiples of three but

we’ve already eliminated that. Four’s been eliminated. We get to five. That’s a prime number. Eliminate all multiples of that. Six has been eliminated. You get to seven, eliminate

all multiples of that. Nice little crisscross pattern. Then up to 11. And you only need to–

when doing the sieve, go to the square

root of the number that you’re seiving which

is root of 100 which is 10. So we’ve got to 10

already so therefore we can guarantee that all of those

numbers in their are primes. And there are few

interesting things to say. First is that all prime numbers

are either one or one less or one more than

a multiple of six because that can only be in

line one or in line five. If this went on to infinity,

it would be the same. But also– which is why

people are kind of fascinated by prime numbers– is that

before you get to them, there’s no way of knowing if

a number is going to be prim or not. They appear to be

kind of randomly sprinkled around

the number line. And that is one of the

reasons why they are still one of the first things

to be studied in numbers and there’s still–

lots of things about them are just not known. Lots of books about

prime numbers. So in 1963, Stanislav

Ulam, who was one of the great mathematicians

of the 20th century, was doodling, got

bored in a math lecture and started to doodle. And he did a great but he

put the one in the middle and then started to

spiral and then started to cross off the prime numbers. And he noticed

something that no one had noticed before–

that prime numbers tend to sit on lines together. And so this is something

after several thousand years no one had ever

bothered to do this. And this is up to 100,

which is the numbers that we saw on the grid before. But if we do it up to 20,000,

you see it a bit more. There are these lines that

prime numbers seem to sit on. And it’s called an Ulam spiral. And really, in maths, you

kind of have total order and that’s a bit boring. You have total chaos. That’s a bit complicated. When you have this sort of in

the bit between order and chaos which you have here is where

things get really interesting. So this is a really kind of

simple yet incredibly deep image of where are

all these primes and what relations do

they have to each other? This is Ulam right

here on the right. And he’s sitting. This is just after

the Second World War near Los Alamos with John

Von Neumann on the left, who I hope that Google has a

shrine to John Von Neumann somewhere because basically

he invented the computer with Alan Turing of course. But a lot of the kind

of conceptual framework of computing was basically

invented by John Von Neumann. In the middle is

Richard Feynmann, the famous American physicist. This is an amazing picture

because you could quite easily argue that these three

men sort of changed the 20th sort of

more than anyone else because they were

instrumental in creating nuclear weapons, the computer,

Von Neumann– possibly the greatest mathematical

genius of the 20th century. Whatever he tried to do he

invented like a new field. And it’s one of the fields

that he and Ulam invented I’m just going to finish

off talking about now which hopefully is right up your

street because it talks about essentially the first computer

craze that there ever was. So Von Neumann was

inventing so much stuff. And it was in the

1940s, early ’50s, started to worry about

what he was creating. Computers, robots, what’s

going to happen to robots? What would it take, he asked,

for a robot or a machine to replicate itself? And this is actually a kind

of psychologic problems. It’s like a

mathematical problem. It’s not really a

biological problem. It’s a kind of conceptual

idea because just say you’ve got the

machine and you’ve got, say, the instructions. If you put the instructions

in the machine, can the machine make itself

from the instructions? Because the

instructions, obviously, are part of the machines. So you cannot replicate. And basically it can’t

in a finite world. Just say you’ve got a machine

with a set of instructions. What you want to do is

create exactly that. A machine and a set

of instructions. Just say the machine

reads the instructions and it’s got instructions

how to rebuild a machine. Does that set of instructions

have instructions on how to build a new

set of instructions because if it does, then

that set of instructions within the instructions. It’s got to have some

other instructions. And then you get this

infinite regress. It’s a finite system. It can’t happen. So he realized that

if you have a machine with a set of instructions

to self-replicate, you need to treat the

instructions in two ways. You need to read it

to build the machine and then you need to

duplicate it and then present it to the new machine. So there are two

ways– basically, you need these two elements. And when, a few years

ago, Watson and Crick discovered DNA, they realized

that essentially the same thing happens in humans

more in general terms. The DNA is the instructions. It has instructions

for the cell. But DNA does not contain

instructions for DNA because DNA replicates. It’s a double helix and one

of the helixes splits off and then you create another one. So this is another fantastic

example of something which the mathematics was

conceptually discovered years before it was shown

to be that was the way– it’s like

the Higgs boson. It was kind of

mathematically proved and then, decades later,

it was discovered. What John Von Neumann

wanted to do– he’s also like a fantastic

entertaining character. He loved parties and

sometimes would hold parties just because he liked working

to the sound of parties. So he would be there

kind of mixing together and then go back to his room

and do some amazing maths. He wanted to think

about building a self-replicating

robot, something which can use instructions,

then duplicate them. But it was just too complicated

actually working out how to build it

in the real world so Ulam who, as we’ve seen,

loves grids and things like that and

patterns, said, let’s invent a new kind of

mathematical abstraction. Which they did, called

the cellular automaton. And a cellular automaton

essentially is a grid of cells. The cells can have

different states but the behavior of

the cells depends only on neighboring cells. I’m going to explain now a

very simple cellular automaton called the Game of Life

because Von Neumann invented it to try and find something

that would self-replicate and he managed to do

that conceptually. And then, a decade or

so later, at the end of the ’60s, a British

mathematician called John Conway, who’s

still alive– he’s at Princeton at the

moment– invented what is one of the most famous

examples of what’s called recreational maths but

it’s also quite serious called the Game of Life. It’s not a game. It’s a cellular automaton. Life because it emulates

evolution of life. So this is the grid. The game of Life– I’m going

to explain how it works. All of it takes place on

a two-dimensional grid. The grid can be infinite, and

it has these orthogonal cells. The cells can have two

states, either alive or dead. This is a live cell. All around it are dead. And it’s immediate

neighbors– there are eight of them– are these

ones here all around it. So its behavior depends only

on those eight cells around it. And these are the

rules that it obeys. There are the Game

of Life genetic laws. One, if you have a live, if

it’s surrounded by zero or one live neighbors, it

dies by loneliness. If it has two or three live

neighbors, it survives. Four or more live neighbors,

it dies by overcrowding. Really, really simple. Basically, it a live cell is

surrounded by two or three, it survives. If it’s not, it dies. A dead cell, if it has

exactly three live neighbors, it becomes alive. And in all other

cases, it stays dead. So Conway– this

was the late ’60s– he had one of those grids. It was basically a Go board. There were no computers then. And he used Go counters just

to see what would happen. So let’s see what happens. This is a pattern

with three shapes. What the Game of Life does

is what we will always do– we will start

with a pattern. And the idea is to put

an interesting pattern on the board and then you

don’t, in fact, touch it at all. It then just

evolves and changes. So what we want to

see is how it evolves. And Conway chose

his rules to make the most unpredictable

and interesting way that things evolve. So what we’re going to do here–

let’s look at this one here, surrounded by one live

neighbor, it will die. Surrounded by two live

neighbors, it will survive. Surrounded by one live

neighbor, it will die. A dead cell surrounded by

three, it will survive. So what happens is

that you work out how each one is going

to behave and then you apply the rules on

everything and then you get a new generation. So what would happen here? The new generation

would be that. OK. And then it dies because

there are only two left. This one here. That becomes that. And then it dies. This one here. We’re going with shapes

with three live cells. Becomes that. And then it just stays there. It’s called a stable form. So at the beginning, they

were like zoologists. They’ve got this new world. They were just

trying to work out what was there, finding the

patterns and what would happen. This one here, another

three cell pattern. Becomes that, then

that, and then that. That’s known as the blinker. OK. Not that interesting yet. When you start to add more,

most things end up dying, but sometimes you get

amazing bursts of life. So this four-cell pattern

becomes this then this then this then this. It’s four blinkers, which

is called traffic lights. Let’s go and look at

a five cell pattern. So they had done four-cell,

see what happened to them. They were cataloging them. And then they noticed

something amazing. And this is when basically

the study of life really took off because look

at this five-star pattern. It repeats. It’s called a glider. Every four generations,

it’s moved one space down and one cell along on the grid. So essentially it moves. It looks like it’s moving. And this was the– Conway

called these spaceships in the kind of etymology of the

kind of animals or the living things that you get. And he wondered and

in Scientific American he said, does anyone

know if there’s any pattern that can grow and

have more and more live cells. Because this one it all stays

at five cells all the way down. Something that can grow

and get infinitely big. And he put a reward in

Scientific American. And the Martin Gardener

article that wrote about it was the most read

article he ever wrote. And this is a guy

who is like the guru of all kind of popularization

of maths and science. And this was Conway in

Cambridge writing about it. Martin Gardener in

America writing about it. But the real development

started to happen at MIT. The Game of Life

came along at a time where computing

was just beginning. But you would only

have computers if you were in a big

kind of organization or if you were at a

university like MIT. And I don’t know

if you know where the word “hacking” comes from. The word “hacking” originally

comes from the MIT Model Railway Club, and a hack is when

you kind of customize something just for the hell of

it, just to make it fun. And so the hackers

were the people who worked at the MIT

Model Railway Club. But when computers

came along, they became not so interested

in model railways and worked on computers. And the thing with

Game of Life, as you saw– incredibly simple

rules but incredibly complex behavior. It became basically the

first computer craze that the hackers

at MIT– and this is essentially

the people whose– the forefathers or forebrothers,

say, of Steve Jobs, Bill Gates, et cetera, were

the hackers at MIT working on the Game of Life, which

was basically the biggest deal in computing

in the early ’70s. And Bill Gosper, who was the

kind of king of the hackers, discovered and won the prize

for the first pattern that grows without bound,

which is this. And it looks like a

kind of pair of lungs. And what it’s doing is

called a glider gun. And basically it just, every

period, spits out a glider. And that will carry

on, getting bigger. And the total of live cells

just gets bigger and bigger. And the more that

people looked at this, they started to find

other patterns that did other interesting things. So this thing here

is called an eater. And what an eater

does is an eater will eat anything

that you send at it. So this gave the

first indication that this might actually

have some use because it’s like how to build

self-repairing object. But also it meant that

you could start to just have these amazing kind

of engineering patterns where you could build

massive patterns and all the debris you could

have eater just clean them up. So let’s have a look at

the eater, what happens. So this is a glider coming in

it to an eater, gets eaten. This is another spaceship,

just gets eaten. And let’s now look at–

this is the sort of thing that people started to build. So this is a glider moving. And I’m going to

be going out of it. What’s happening? So it’s going to this

other massive pattern. What’s going to

happen to the glider? Bing. It bounces. So that’s basically

a glider reflector. It will bounce a

glider at 90 degrees. So as we keep on

going out we see we’ve got these gliders

in kind of outer space. And what’s interesting is

that now you look at it, you’re thinking it’s like these

little ants walking around. The grid, every cell, is only

responding to the eight cells around it. So there is no kind of

overall guiding thing, but yet there is. So Game of Life is

such a great way to understand how things

at different scales take on different appearances. So we keep on going

out of this pattern. All these are

gliders in streams. And here we are. In fact, what this

is– it’s a gun. It goes back in in

a second, magnifies. These are the two

kind of barrels. And everything is working

in such synchronicity that– going in. This is a spaceship, a special

slightly larger spaceship than a glider. That will be shot out. So this thing will keep

on going forever, just shooting out those spaceships. Now, what kind of things can you

build using the Game of Life? And I wanted to show

you this pattern. So this is a pattern,

20,000 cells. It’s the sieve of Eratosthenes. What it’s doing– it’s a gun

that’s shooting out spaceships only in the prime

number positions, OK? This is two, three. Four’s not prime

so it’s not there. That’s Five. Six is not there. That’s seven. That’s nine. Nine’s not going to come back

because a glider gets shot at it. OK. So what is actually

happening here is that, as that

goes along there, this thing here shoots

a glider every three. In other words, anything

that’s divisible by three it’s going to kill. This shoots something every

five, this every seven, this every nine. And if I go to the

next, this one here. Hang on. Yeah. So again. So this shooting every three. That’s seven. This is nine. Because we know all even

numbers aren’t prime, we only shoot at the odds. That is nine. That’s 11. That’s going to get through. So it’s the sieve

of Eratosthenes turned into this kind of

intergalactic shootout between gliders and spaceships. That’s 11. The next one is 15 so 15

is divisible by that’s the three coming in. That’s the five coming in. That’s 15. That’s going to get shot. So we could let this

pattern go forever and it would just

shoot out gliders. I’ve one slide to go left,

which is this one here. So what is the extent

of the Game of Life? What can the Game of Life do? It can find prime numbers. Actually, speeding up so we have

enough time, the Game of Life can do absolutely anything

that any computer can do. Right? Because what is a computer

at its most basic level? It’s a bunch of wires with

a pulse going through it. It’s binary. You can emulate a wire with

a pulse going through it by a stream of

gliders like here. So when there’s a

space it’s a zero. When there’s a

glider, there’s a one. What else do you need? You need logic gates. You can make the

three logic gates. There are patterns for

the three logic gates. And you need wires to be

able to cross each other. You can do that just by

thinning out the wires so they don’t hit. And you need a memory register. You can do that with a

block, which is the square. And a block at a certain

distance, keeping the memory. So the mathematical

term for that would be that Life is

universal because it’s capable of universal

computation. So even though it would be a

very inefficient way to do it, you could actually

use the Game of Life, which incredibly simple

rules can do anything that the most complicated,

sophisticated computer could ever do. And I guess on that note,

which is kind of quite a wow. We come to the end of the talk. That’s the name

of the book, which has more information both about

emotional responses to numbers, loads of things in

between, culminating in cellular automata. And the whole idea

about cellular automata is that it’s really good

for emulating replication. And people have now

started to get patterns that do self-replicate in

the sort of Von Neumann sense of self-replicating. And mathematicians there’s

only a matter of time that you will be able to create

self-replicating creatures that live on a kind of big infinite

grid of the Game of Life. So the kind of game

is turning into life. And there are philosophers who

say that, actually, the best explanation for the

world and the universe is a bit like the Game

of Life– that there’s some initial

configuration of kind of discrete elements of some way

and that we are just squillions of generations beyond

that beginning grid. So, again, it’s interesting

to think philosophically about that. That’s the first

book, “Numberland.” That’s the football

book that I did. I blog, “Alex’s Adventures in

Numberland,” at the “Guardian” every couple of weeks. And I’m on social

media if you want to follow any of the maths

popularization work that I do. Thank you very much. Sorry for talking for

a bit longer than I probably expected to. Thank you. PETER: Thank you so much. That was crazy stuff. Brilliant. And people have to

run, I’m afraid. It’s not rudeness. I think people have

meeting at 2:00. But I think we’ve probably

run out of time for questions. Except you neglected to say

what your favorite number was. ALEX BELLOS: Well,

that’s the last thing. I don’t have a

favorite number which is why I got interested

in it in the first place. I don’t have one. PETER: He’ll be around

for a few minutes. ALEX BELLOS: I will

totally be around. PETER: If you want to

ask questions one-to-one, feel free to do so. Thank you very much indeed. [APPLAUSE]